Polar leapfrog filter

ABSTRACT

A polar leapfrog filter includes at least one polar network. The polar network comprises a differentiator constituted by an operational amplifier having input and output terminals, and a first integrator formed by a first capacitor for providing negative feedback to the operational amplifier and a first variable transconductance amplifier; and a second integrator formed by a second capacitor for providing negative feedback to the first integrator, and a second variable transconductance amplifier. In the case where two or more said polar networks are incorporated, an integrator is provided between adjacent ones of the polar networks. The total number of all the circuits is selected to be odd and equal to the order of the filter. The adjacent ones of the circuits are connected in such a manner that leapfrog type negative feedback is effected.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a polar leapfrog filter which can beconstructed in the form of an active filter, and more particularly itpertains to an odd-order polar leapfrog low-pass filter.

2. Description of the Prior Art

Heretofore, such passive filters as shown in FIGS. 1 and 2 have beenwidely used. However, it is the recent trend that so-called activefilters are employed in lieu of such passive filters, as the result ofperipheral circuits associated therewith being constructed in the formof semiconductor integrated circuit. Generally, an active filter is madeup of components, each of which comprises a resistor, a capacitor, andan operational amplifier, and constructed in the form of a sallen-keycircuit, a biquad circuit or an FDNR (frequency-dependent negativeresistance) circuit by combining such operational amplifiers.Alternatively, a desired filter is constructed by using such circuits asunits. In an attempt to change the filter characteristics, with thebiquad circuit or the like, it is required that the constants for theresistors and capacitors be changed. With the FDNR filter, on the otherhand, difficulties are experienced in an attempt to adjust the filtercharacteristics thereof since it is the usual practice that several suchfilters are interconnected with each other and it is required thatconstants for the elements of each such filter be changed. The othertypes of filter use variable resistors to make variable the filtercharacteristics thereof; thus, such filters are constructed inevitablyin the form of a hybrid integrated circuit. Alternatively, it isrequired that chip components of a predetermined resistance value bepre-selected and mounted onto a printed circuit board, whichdisadvantageously leads to a increase in the size of the filter. Ineither case, such filters are disadvantageous in that they cannot beconstructed in the form of a monolithic integrated circuit since thevariable resistors or the pre-selected chip components should be mountedonto the printed circuit board as mentioned above.

In an attempt to make a Cauer filter or the like or achieve desiredfilter characteristics, it is required that the filter be constructed inthe form of polar type having a damping pole, i.e., transmission zeropoint (pole zero) at a definite frequency. With the above-mentionedconventional arrangements, however, when it is attempted to achievethis, problems arise in that a number of parts are required so that thecircuit arrangement turns out to be complicated and difficulties areencountered in an attempt to achieve an odd-order arrangement, though aneven-order arrangement is achievable, as is the case with a biquadcircuit. The problems with a filter constructed in the form of a hybridintegrated circuit having a number of parts mounted on a printed circuitboard, are such that most such filters are bulky and difficult to adjustthe filter characteristics thereof.

SUMMARY OF THE INVENTION

It is an object of the present invention to provide a polar leapfrogfilter constructed in accordance with the leapfrog simulation procedure(refer to M. E. Van Valkenburg: "Analog Filter Design", 1982, CBSCollege Publishing), thereby eliminating the above-described drawbacksof the prior art.

The present invention employs the leapfrog simulation procedure asmentioned just above, and it is also based on the invention disclosed inU.S. patent application Ser. No. 798,215 filed Nov. 26, 1991, now U.S.Pat. No. 5,177,382, corresponding to Japanese Patent Application No.333136/1990 filed Nov. 29, 1990.

An important advantage of the present invention is such that a Cauerfilter can be very easily achieved according to the present invention.Another important advantage is such that a higher order filter can beeasily constructed. Still another important advantage is such that thefilter according to the present invention can readily be constructed inthe form of semiconductor integrated circuit since the main elements ofthe filter are integrators, so that the number of parts thereof as wellas the size thereof can be reduced.

Furthermore, the polar leapfrog filter according to the presentinvention is advantageous in that since the main constitutional elementsthereof are integrators each comprising a variable transconductanceamplifier, the pass bandwidth thereof can be easily adjusted byadjusting current supplied to the differential transistor pair of suchvariable transconductance amplifier to cause the internal resistance ofsuch transistor pair to be changed.

Other objects, features and advantages of the present invention willbecome apparent from the ensuing description taken in conjunction withthe accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates an example of conventional odd-order passive low-passfilter.

FIG. 2 illustrates another example of conventional odd-order passivelow-pass filter.

FIG. 3 is a circuit diagram showing an example of polar network which isused as a constitutional element of the polar leapfrog filter accordingto the present invention.

FIGS. 4A to 4D are block diagrams useful for explaining about the polarnetwork of FIG. 3.

FIG. 5 illustrates the frequency characteristics of the polar networkshown in FIG. 3.

FIG. 6A illustrates an arrangement achieved by scaling the arrangementof FIG. 1.

FIG. 6B illustrates the arrangement of FIG. 6A as represented by usingadmittance and impedance.

FIG. 7A illustrates signal flow developed in accordance with theleapfrog simulation procedure.

FIG. 7B illustrates an arrangement transformed from that of FIG. 7A.

FIG. 7C illustrates an arrangement transformed from that of FIG. 7B.

FIG. 8 is a circuit diagram showing the polar leapfrog filter accordingto an embodiment of the present invention which is constructed bytransforming the passive filter shown in FIG. 1.

FIG. 9 illustrates the arrangement of FIG. 2 as subjected to scaling andrepresented by using admittance and impedance.

FIG. 10A illustrates signal flow developed in accordance with theleapfrog simulation procedure.

FIG. 10B illustrates an arrangement transformed from that of FIG. 10A.

FIG. 10C illustrates an arrangement transformed from that of FIG. 10B.

FIG. 11 is a circuit diagram showing the polar leapfrog filter accordingto another embodiment which is constructed by transforming the passivefilter shown in FIG. 2.

FIG. 12 illustrates the frequency characteristics of the embodimentsshown in FIGS. 8 and 11.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

Description will now be made of embodiments of the present inventionwith reference to FIGS. 3 to 12.

Referring to FIG. 3, an input terminal 1 is connected to a non-invertinginput terminal of an operational amplifier A₁ of a substantiallyinfinite gain, and an output terminal 2 is led out of the outputterminal of the operational amplifier A₁. The output terminal of theoperational amplifier A₁ is connected to a non-inverting input terminalof a variable transconductance amplifier A₂, the output terminal ofwhich is coupled to an inverting input terminal of the operationalamplifier A₁ and grounded through a capacitor C₁. The variabletransconductance amplifier A₂ and the capacitance C₁ constitute anintegrator 11 which is arranged to provide negative feedback to theoperational amplifier A₁. The operational amplifier A₁ and theintegrator 11 constitute a differentiator 13. Further, the outputterminal of the variable transconductance amplifier A₂ is connected to anon-inverting input terminal of another variable transconductanceamplifier A₃, the output terminal of which is grounded through a secondcapacitor C₂ and connected to an inverting input terminal of thevariable transconductance amplifier A₂. A second integrator 12 isconstituted by the variable transconductance amplifier A₃ and thecapacitor C₂. An inverting input terminal of the variabletransconductance amplifier A₃ is grounded.

As will be appreciated from the above explanation, the circuit of FIG. 3is a negative feedback circuit comprising in combination the integrator12 and differentiator 13 and which represents a damping pole (pole zero)at a predetermined frequency as will be seen from FIG. 5. Thus, thecircuit of FIG. 3 comprises a polar type circuit having a damping pole,which will be referred to as "polar network" hereinafter.

The polar network of FIG. 3 has a damping pole which occurs at a pointwhere the filter characteristics (a) and (b) of the integrator 12 anddifferentiator 13 as combined cross each other, as shown in FIG. 5. Itis possible to shift the damping pole by adjusting the operatingcurrents of the variable transconductance amplifiers A₂ and A₃ to set upthe transconductances gm₁ and gm₂ of variable transconductanceamplifiers A₂ and A₃. By adjusting the operating current of the variabletransconductance amplifier A₃ constituting the integrator 12, forexample, it is possible to cause the damping pole to be shifted from P₁at a frequency f₀ to P₂ at a frequency f₁. Further, by adjusting theoperating currents of the variable transconductance amplifiers A₂ and A₃constituting the integrators 11 and 12 respectively at the same time andin the same direction, it is possible to change the damping quantityalone, while keeping the damping pole at the predetermined frequency f₀.

The block diagram of FIG. 4D corresponds to the polar network of FIG. 3;thus, it will be explained that the network of FIG. 3 can be converteddirectly from the block diagram of FIG. 4D. As mentioned above, thepolar network of FIG. 3 is a negative feedback circuit comprising theintegrator 12 and differentiator 13 in combination; thus, the transferfunction of the polar network shown in FIG. 3 is equivalent to a sum ofthe transfer functions of the integrator 12 and differentiator 13. Inthis way, assuming that the transfer functions of the differentiator andintegrator are represented by sC₁ /gm₁ and gm₂ /sC₂ respectively, thetransfer function of the polar network shown in FIG. 3 is given asfollows:

    V.sub.2 /V.sub.1 =sC.sub.1 /gm.sub.1 +gm.sub.2 /sC.sub.2   (1)

where V₁ is an input voltage, V₂ is an output voltage, and gm₁ and gm₂are the transconductances of the variable transconductance amplifiers A₂and A₃ respectively.

FIG. 4A is a block diagram illustrating a common negative feedbackcircuit comprising an adder 3, and blocks 4 and 5. FIG. 4B is a blockdiagram wherein an integrator having a predetermined transfer functionis provided in each of blocks 7 and 8 which correspond to the blocks 4and 5 of FIG. 4A respectively. FIG. 4C illustrates an arrangement inwhich blocks 10 and 14 corresponding to the blocks 4 and 5 of FIG. 4Aare constituted by an operational amplifier having an infinite gain anda block having a predetermined transfer function, respectively. FIG. 4Dis a block diagram showing the above-mentioned polar network which isconstituted by the blocks of FIGS. 4B and 4C. The polar network of FIG.3 can be realized through direct transformation of the blocks of FIG.4D.

The transfer function given by the equation (1) will now be sought. Thetransfer function of the arrangement shown in FIG. 4A, which comprisesthe blocks of a negative feedback circuit called basic feedback system,is given by equation (2) as follows:

    (V.sub.1 -αV.sub.2)β=V.sub.2                    (2)

where α is the variable of the block 4, and β is the variable of theblock 5.

Thus, from the equation (2), the transfer function of the block diagramshown in FIG. 4A is given as follows:

    V.sub.2 /V.sub.1 =1/(α+1/β)                     (3)

In the block diagram of FIG. 4A, let it be assumed that the variables αand β are substituted with integrator transfer functions as shown below.

    α=gm.sub.2 /sC.sub.2

    β=gm.sub.1 /sC.sub.1                                  (4)

Then, the block diagram of FIG. 4A can be transformed to the blockdiagram of FIG. 4B comprising blocks 7 and 8. Thus, by substituting theequation (4) for the equation (3), the transfer function can beexpressed as follows:

    V.sub.2 /V.sub.1 =1/(sC.sub.1 /gm.sub.1 +gm.sub.2 /sC.sub.2)(5)

Further, the variables α and β of the blocks 4 and 5 in the blockdiagram of FIG. 4A are rewritten as follows:

    α=1/(sC.sub.1 /gm.sub.1 +gm.sub.2 /sC.sub.2)

    β=∞                                             (6)

The resulting block diagram turns out to be as shown in FIG. 4C, thetransfer function of which is given as follows, by substituting theequations (6) for the variables α and β in the equation (3):

    V.sub.2 /V.sub.1 =sC.sub.1 /gm.sub.1 +gm.sub.2 /sC.sub.2   (7)

The equation (7) indicates that the block diagram of FIG. 4C provides atransfer function which is equivalent to one obtained by adding up thecharacteristics of the differentiator and integrator such as representedby the equation (1).

The transfer function of the block 14 shown in FIG. 4C is given as1/(sC₁ /gm₁ +gm₂ /sC₂), which is identical with the transfer functionrepresented by the equation (5).

Thus, by combining the block diagrams of FIGS. 4B and 4C, the transferfunction of the equation (7) can be illustrated as in the block diagramof FIG. 4D. In this way, it has been found that the block diagram ofFIG. 4D can be transformed to the circuit arrangement of FIG. 3.

As will be appreciated from the above discussion, according to thepresent invention, an odd-order polar leapfrog filter is constructed onthe basis of the matters explained hereinabove with reference to FIGS. 3to 5. The odd-order leapfrog filter is arranged such that in the casewhere at least one said polar network is incorporated therein, anintegrator is provided at each of the input and output portions thereof;in the case where two or more said polar networks are incorporatedtherein, an integrator is provided between said polar networks, thenumber of all the circuits being selected to be odd and equal to theorder of the filter; and leapfrog type negative feedback is providedbetween respective adjacent ones of the above-mentioned circuits.

Referring to FIG. 1, there is shown a conventional odd-order passive LPF(low-pass filter) having no terminal resistor, where n is an odd numberwhich is equal to or greater than 3. Referring to FIG. 8, there isillustrated the polar leapfrog filter according to an embodiment of thepresent invention, which is achieved by transforming the passive filterof FIG. 1 to an active filter in accordance with the leapfrog simulationprocedure mentioned in the preamble portion of the presentspecification.

Description will now be made of the process of designing the activefilter of FIG. 8, which is equivalent to the passive filter of FIG. 1,by using the leapfrog simulation procedure.

In FIG. 1, the element values and frequency are subjected to scaling sothat the value for R₀ ' and the cut-off frequency ω₀ become equal to 1and 1 rad/sec respectively. Assuming that the values for the resistor,capacitor and coil, after having been subjected to the scaling, are R",C" and L" respectively, these values are given as follows (such valueshave no unit):

    R"=R'/R.sub.0 '[-]

    C"=ω.sub.0 C'R.sub.0 '[-]

    L"=ω.sub.0 L'/R.sub.0 '[-]                           (8)

With the coil being regarded as identical with the capacitor, theelement values and cut-off frequency are again subjected to scaling sothat the the value for R" becomes equal to 1Ω and the cut-off frequencybecomes the original value. Assuming that the values for the resistor,capacitor and coil, after having been subjected to scaling, are R'", C'"and L'" respectively, such values are given as follows:

    R'"=R"·1=R'/R.sub.0 '[Ω]

    C'"=C"/ω.sub.0 =C'/R.sub.0 '[F]

    L'"=L"/ω.sub.0 =L'/R.sub.0 '=(L'/R.sub.0 '.sup.2)R.sub.0 '[F](9)

By letting R₀ '=1/gm, the equations (9) can be rewritten as follows:

    R'"=R'gm[Ω]

    C'"=C'/gm[F]

    L'"=(L'gm.sup.2)/gm[F]                                     (10)

As a result of the above successive scaling procedures, the elementvalues in FIG. 1 are transformed as shown in FIG. 6A wherein therespective element values are given by equations (11a) to (11f)respectively. In this case, gm=1/R₀ '.

    C.sub.1 =C.sub.1 '[F]

    C.sub.i-1 =C.sub.i-1 '[F]

    L.sub.i-1 =L.sub.i-1 'gm.sup.2 [F]

    C.sub.n-1 =C.sub.n-1 '[F]

    L.sub.n-1 =L.sub.n-1 'gm.sup.2 [F]

    C.sub.n =C.sub.n '[F]                                      (11)

In FIG. 6A, by representing the floating elements in the form ofadmittance Y and the grounded elements in the form of impedance Z, FIG.6A can be transformed to FIG. 6B. The respective admittance andimpedance values in FIG. 6B are given as shown by equations (12) asfollows:

    Y.sub.0 =1

    Z.sub.1 =gm/sC.sub.1

    Y.sub.i-1 =sC.sub.i-1 /gm+gm/sL.sub.i-1

    Z.sub.i =gm/sC.sub.i

    Y.sub.n-1 =sC.sub.n-1 /gm+gm/sL.sub.n-1

    Z.sub.n =gm/sC.sub.n                                       (12)

In the equations (12), each admittance and impedance can be regarded asconstituting a transfer function; thus, the transfer functions can beexpressed as indicated by equations (13) as follows:

    T.sub.0 =1

    T.sub.1 =gm/sC.sub.1

    T.sub.i-1 =sC.sub.i-1 /gm+gm/sL.sub.i-1

    T.sub.i =gm/sC.sub.i

    T.sub.n-1 =sC.sub.n-1 /gm+gm/sL.sub.n-1

    T.sub.n =gm/sC.sub.n                                       (13)

The arrangement of FIG. 6B can be developed into such a signal flow asshown in FIG. 7A, in accordance with the leapfrog simulation proceduredisclosed in the "Analog Filter Design" cited in the preamble portion ofthe present specification. In FIG. 7A, the characters in the respectiveblocks represent transfer functions corresponding to the equations (13)respectively, and the circles indicate adders.

By putting the equations (13) in the respective blocks of FIG. 7A, thearrangement of FIG. 7A is transformed to that of FIG. 7B. Recall thatthe block 14 can be rewritten as shown in FIG. 4D as mentionedhereinbefore; by putting such relationship in FIG. 7B, the arrangementof FIG. 7B can be transformed to that of FIG. 7C.

Furthermore, from the fact the polar network of FIG. 3 can be achievedthrough direct transformation from the block diagram of FIG. 4D, it ispossible to construct such a circuit arrangement as shown in FIG. 8, byputting the relationship between FIG. 4D and FIG. 3 in the arrangementof FIG. 7C in which the block 20 is a self feedback type integrator: theblocks 21 and 23 are polar networks; the blocks 22 and 24 areintegrators; and leapfrog type negative feedback is provided to each ofthe blocks except for the block 24. The circuit arrangement of FIG. 7Cconstitutes the odd-order polar leapfrog filter according to theembodiment of the present invention, which is equivalent to thearrangement of FIG. 1 as will readily be appreciated from the abovediscussion. The circuits indicated at 12 and 13 in FIG. 8 correspond tothe integrator 12 and differentiator 13 of FIG. 3, respectively; thus,it will be readily apparent that the circuits 21 and 23, each of whichcomprises the integrator 12 and differentiator 13, correspond to thepolar networks described above with reference to FIG. 3. In FIG. 8,reference numerals 20, 22 and 24 indicate integrators comprisingvariable transconductance amplifiers G₁, G_(i), and G_(n) having theoutput terminals thereof grounded through capacitors C₁, C_(i) and C_(n)respectively. Each of the integrators 20, 22 and 24 corresponds to theintegrator 12, and only the integrator 20 is constructed in the form ofself negative feedback type. Furthermore, negative feedback is providedin the form of leapfrog from the output side integrator 24 to theimmediately preceding polar network 23, from the polar network 23 to theimmediately preceding integrator 22, from the integrator 22 to theimmediately preceding polar network 21, and from the polar network 21 tothe integrator 20 at the input. In this case, the sum of the number ofthe integrators and the number of the polar networks is odd and equal tothe order of the filter, and it is possible to increase the order of thepolar leapfrog filter by increasing the number of the combinations ofthe polar network (such as 21, 23) and integrator (such as 22, 24) whichare connected to each other as indicated by dotted lines.

Referring to FIG. 2, there is shown a conventional odd-order passive LPF(low-pass filter) having a terminal resistor, which is identical withthe arrangement of FIG. 1 except that the terminal resistor is provided.FIG. 11 illustrates the polar leapfrog filter according to anotherembodiment of the present invention, which is achieved by transformingthe passive filter of FIG. 2, like that of FIG. 1, to an active filterin accordance with the leapfrog simulation procedure.

The arrangement of FIG. 2 is transformed to such an arrangement as shownin FIG. 9, by carrying out the scaling procedure and then representingthe floating elements by admittance Y and the grounded elements byimpedance Z as in the case of FIG. 1. In this case, too, the admittanceand impedance can be regarded as transfer functions; thus, the transferfunctions are given by equations (14) as follows:

    T.sub.0 =1

    T.sub.1 =gm/sC.sub.1

    T.sub.i-1 =sC.sub.i-1 /gm+gm/sL.sub.i-1

    T.sub.i =gm/sC.sub.i

    T.sub.n-1 =sC.sub.n-1 /gm+gm/sL.sub.n-1

    T.sub.n =gm/(sC.sub.n +gm)                                 (14)

The arrangement of FIG. 9 can be developed into such a signal flow asshown in FIG. 10A, in accordance with the leapfrog simulation proceduredisclosed in the "Analog Filter Design" cited in the preamble portion ofthe present specification. In FIG. 10A, the characters indicated in therespective blocks correspond to the transfer functions represented bythe equations (14) respectively, and the circles represent adders.

By putting the equations (14) in the respective blocks of FIG. 10A, thearrangement of FIG. 10A is transformed to that of FIG. 10B. Recall thatthe arrangement of FIG. 4A can be transformed to the arrangement of FIG.4D as mentioned hereinbefore; by putting such relationship in FIG. 10B,the arrangement of FIG. 10B can be transformed to that of FIG. 10C.

As in the case of the embodiment shown in FIG. 8, the arrangement ofFIG. 10C can be transformed to the arrangement of FIG. 11 constitutingthe odd-order polar leapfrog filter according to the second embodimentof the present invention which is equivalent to the arrangement of FIG.2. The embodiment of FIG. 11 is different from the embodiment of FIG. 8only in that the output side integrator 24 is constructed as selfnegative feedback type, which corresponds to the fact that the filter ofFIG. 2 has a terminal resistor.

FIG. 12 illustrates the frequency characteristics of the polar leapfrogfilters shown in FIGS. 8 and 11, pass band of which is changed asindicated by (a), (b) and (c) as the transconductance gm of the variabletransconductance amplifier constituting the integrator is changed to 1,0.5 and 2 mS (Simens or Mho) by changing the operating current flowingthrough the differential transistor pair of the variabletransconductance amplifier.

While the present invention has been illustrated and described withrespect to specific embodiments thereof, it is to be understood that thepresent invention is by no means limited thereto but encompasses allchanges and modifications which will become possible within the scope ofthe appended claims.

We claim:
 1. A polar leapfrog filter including a polar network whichcomprises a differentiator constituted by an operational amplifierhaving input and output terminals, and a first integrator for providingnegative feedback to said operational amplifier, said first integratorcomprising a first capacitor and a first variable transconductanceamplifier; and a second integrator for providing negative feedback tosaid first integrator, said second integrator comprising a secondcapacitor, and a second variable transconductance amplifier.
 2. A polarleapfrog filter according to claim 1, wherein the integrator isconstructed at the input side of said polar network.
 3. A polar leapfrogfilter according to claim 1, wherein both of the integrators areconstructed at the input side and output side of said polar network. 4.A polar leapfrog filter including at least one polar network whichcomprises a differentiator constituted by an operational amplifierhaving input and output terminals, and a first integrator for providingnegative feedback to said operational amplifier, said first integratorcomprising a first capacitor and a first variable transconductanceamplifier; and a second integrator for providing negative feedback tosaid first integrator, said second integrator comprising a secondcapacitor and a second variable transconductance amplifier, andintegrators provided at the input and output sides of said filterrespectively, wherein the total number of the circuits is selected toprovide an odd order and equal order for the filter, and adjacent onesof the circuits are connected with each other in such a manner thatleapfrog type negative feedback is effected.
 5. A polar leapfrog filterincluding a plurality of polar networks, each of which comprises adifferentiator constituted by an operational amplifier having input andoutput terminals, and a first integrator for providing negative feedbackto said operational amplifier, said first integrator comprising a firstcapacitor and a first variable transconductance amplifier; and a secondintegrator for providing negative feedback to said first integrator,said second integrator comprising a second capacitor and a secondvariable transconductance amplifier; integrators provided at the inputand output sides of said filter respectively; and a further integratorprovided between said polar networks, wherein the total number of thecircuits is selected to provide an odd order and equal order for thefilter, and adjacent ones of the circuits are connected with each otherin such a manner that leapfrog type negative feedback is effected.
 6. Apolar leapfrog filter according to claim 5, wherein the integratorprovided at the input side is constructed in the form of self negativefeedback type.
 7. A polar leapfrog filter according to claim 5, whereinboth of the integrators provided at the input side and output side areconstructed in the form of self negative feedback type.